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Theorem prfcl 17442
Description: The pairing of functors 𝐹:𝐶𝐷 and 𝐺:𝐶𝐷 is a functor 𝐹, 𝐺⟩:𝐶⟶(𝐷 × 𝐸). (Contributed by Mario Carneiro, 12-Jan-2017.)
Hypotheses
Ref Expression
prfcl.p 𝑃 = (𝐹 ⟨,⟩F 𝐺)
prfcl.t 𝑇 = (𝐷 ×c 𝐸)
prfcl.c (𝜑𝐹 ∈ (𝐶 Func 𝐷))
prfcl.d (𝜑𝐺 ∈ (𝐶 Func 𝐸))
Assertion
Ref Expression
prfcl (𝜑𝑃 ∈ (𝐶 Func 𝑇))

Proof of Theorem prfcl
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prfcl.p . . . 4 𝑃 = (𝐹 ⟨,⟩F 𝐺)
2 eqid 2824 . . . 4 (Base‘𝐶) = (Base‘𝐶)
3 eqid 2824 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
4 prfcl.c . . . 4 (𝜑𝐹 ∈ (𝐶 Func 𝐷))
5 prfcl.d . . . 4 (𝜑𝐺 ∈ (𝐶 Func 𝐸))
61, 2, 3, 4, 5prfval 17438 . . 3 (𝜑𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
7 fvex 6664 . . . . . . 7 (Base‘𝐶) ∈ V
87mptex 6967 . . . . . 6 (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) ∈ V
97, 7mpoex 7760 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) ∈ V
108, 9op1std 7682 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
116, 10syl 17 . . . 4 (𝜑 → (1st𝑃) = (𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
128, 9op2ndd 7683 . . . . 5 (𝑃 = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩ → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
136, 12syl 17 . . . 4 (𝜑 → (2nd𝑃) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)))
1411, 13opeq12d 4792 . . 3 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ = ⟨(𝑥 ∈ (Base‘𝐶) ↦ ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩), (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))⟩)
156, 14eqtr4d 2862 . 2 (𝜑𝑃 = ⟨(1st𝑃), (2nd𝑃)⟩)
16 prfcl.t . . . . 5 𝑇 = (𝐷 ×c 𝐸)
17 eqid 2824 . . . . 5 (Base‘𝐷) = (Base‘𝐷)
18 eqid 2824 . . . . 5 (Base‘𝐸) = (Base‘𝐸)
1916, 17, 18xpcbas 17417 . . . 4 ((Base‘𝐷) × (Base‘𝐸)) = (Base‘𝑇)
20 eqid 2824 . . . 4 (Hom ‘𝑇) = (Hom ‘𝑇)
21 eqid 2824 . . . 4 (Id‘𝐶) = (Id‘𝐶)
22 eqid 2824 . . . 4 (Id‘𝑇) = (Id‘𝑇)
23 eqid 2824 . . . 4 (comp‘𝐶) = (comp‘𝐶)
24 eqid 2824 . . . 4 (comp‘𝑇) = (comp‘𝑇)
25 funcrcl 17122 . . . . . 6 (𝐹 ∈ (𝐶 Func 𝐷) → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
264, 25syl 17 . . . . 5 (𝜑 → (𝐶 ∈ Cat ∧ 𝐷 ∈ Cat))
2726simpld 498 . . . 4 (𝜑𝐶 ∈ Cat)
2826simprd 499 . . . . 5 (𝜑𝐷 ∈ Cat)
29 funcrcl 17122 . . . . . . 7 (𝐺 ∈ (𝐶 Func 𝐸) → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
305, 29syl 17 . . . . . 6 (𝜑 → (𝐶 ∈ Cat ∧ 𝐸 ∈ Cat))
3130simprd 499 . . . . 5 (𝜑𝐸 ∈ Cat)
3216, 28, 31xpccat 17429 . . . 4 (𝜑𝑇 ∈ Cat)
33 relfunc 17121 . . . . . . . . 9 Rel (𝐶 Func 𝐷)
34 1st2ndbr 7724 . . . . . . . . 9 ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
3533, 4, 34sylancr 590 . . . . . . . 8 (𝜑 → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
362, 17, 35funcf1 17125 . . . . . . 7 (𝜑 → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
3736ffvelrnda 6832 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
38 relfunc 17121 . . . . . . . . 9 Rel (𝐶 Func 𝐸)
39 1st2ndbr 7724 . . . . . . . . 9 ((Rel (𝐶 Func 𝐸) ∧ 𝐺 ∈ (𝐶 Func 𝐸)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
4038, 5, 39sylancr 590 . . . . . . . 8 (𝜑 → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
412, 18, 40funcf1 17125 . . . . . . 7 (𝜑 → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
4241ffvelrnda 6832 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
4337, 42opelxpd 5574 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩ ∈ ((Base‘𝐷) × (Base‘𝐸)))
4411, 43fmpt3d 6861 . . . 4 (𝜑 → (1st𝑃):(Base‘𝐶)⟶((Base‘𝐷) × (Base‘𝐸)))
45 eqid 2824 . . . . . 6 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) = (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
46 ovex 7171 . . . . . . 7 (𝑥(Hom ‘𝐶)𝑦) ∈ V
4746mptex 6967 . . . . . 6 ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V
4845, 47fnmpoi 7751 . . . . 5 (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))
4913fneq1d 6427 . . . . 5 (𝜑 → ((2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)) ↔ (𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩)) Fn ((Base‘𝐶) × (Base‘𝐶))))
5048, 49mpbiri 261 . . . 4 (𝜑 → (2nd𝑃) Fn ((Base‘𝐶) × (Base‘𝐶)))
5113oveqd 7155 . . . . . 6 (𝜑 → (𝑥(2nd𝑃)𝑦) = (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦))
5245ovmpt4g 7279 . . . . . . 7 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩) ∈ V) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
5347, 52mp3an3 1447 . . . . . 6 ((𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥(𝑥 ∈ (Base‘𝐶), 𝑦 ∈ (Base‘𝐶) ↦ ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
5451, 53sylan9eq 2879 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦) = ( ∈ (𝑥(Hom ‘𝐶)𝑦) ↦ ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩))
55 eqid 2824 . . . . . . . . 9 (Hom ‘𝐷) = (Hom ‘𝐷)
5635adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
57 simprl 770 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
58 simprr 772 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
592, 3, 55, 56, 57, 58funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
6059ffvelrnda 6832 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐹)𝑦)‘) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
61 eqid 2824 . . . . . . . . 9 (Hom ‘𝐸) = (Hom ‘𝐸)
6240adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
632, 3, 61, 62, 57, 58funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
6463ffvelrnda 6832 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ((𝑥(2nd𝐺)𝑦)‘) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
6560, 64opelxpd 5574 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
664adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐹 ∈ (𝐶 Func 𝐷))
675adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → 𝐺 ∈ (𝐶 Func 𝐸))
681, 2, 3, 66, 67, 57prf1 17439 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
691, 2, 3, 66, 67, 58prf1 17439 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
7068, 69oveq12d 7156 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩))
7137adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
7242adantrr 716 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
7336ffvelrnda 6832 . . . . . . . . . 10 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7473adantrl 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
7541ffvelrnda 6832 . . . . . . . . . 10 ((𝜑𝑦 ∈ (Base‘𝐶)) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7675adantrl 715 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
7716, 17, 18, 55, 61, 71, 72, 74, 76, 20xpchom2 17425 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩(Hom ‘𝑇)⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
7870, 77eqtrd 2859 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
7978adantr 484 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)) = ((((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)) × (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦))))
8065, 79eleqtrrd 2919 . . . . 5 (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) ∧ ∈ (𝑥(Hom ‘𝐶)𝑦)) → ⟨((𝑥(2nd𝐹)𝑦)‘), ((𝑥(2nd𝐺)𝑦)‘)⟩ ∈ (((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
8154, 80fmpt3d 6861 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(2nd𝑃)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝑃)‘𝑥)(Hom ‘𝑇)((1st𝑃)‘𝑦)))
82 eqid 2824 . . . . . . 7 (Id‘𝐷) = (Id‘𝐷)
8335adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
84 simpr 488 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝑥 ∈ (Base‘𝐶))
852, 21, 82, 83, 84funcid 17129 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐷)‘((1st𝐹)‘𝑥)))
86 eqid 2824 . . . . . . 7 (Id‘𝐸) = (Id‘𝐸)
8740adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
882, 21, 86, 87, 84funcid 17129 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝐸)‘((1st𝐺)‘𝑥)))
8985, 88opeq12d 4792 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩ = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
904adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐹 ∈ (𝐶 Func 𝐷))
915adantr 484 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐺 ∈ (𝐶 Func 𝐸))
9227adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐶 ∈ Cat)
932, 3, 21, 92, 84catidcl 16942 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝐶)‘𝑥) ∈ (𝑥(Hom ‘𝐶)𝑥))
941, 2, 3, 90, 91, 84, 84, 93prf2 17441 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ⟨((𝑥(2nd𝐹)𝑥)‘((Id‘𝐶)‘𝑥)), ((𝑥(2nd𝐺)𝑥)‘((Id‘𝐶)‘𝑥))⟩)
951, 2, 3, 90, 91, 84prf1 17439 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
9695fveq2d 6655 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩))
9728adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐷 ∈ Cat)
9831adantr 484 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝐶)) → 𝐸 ∈ Cat)
9916, 97, 98, 17, 18, 82, 86, 22, 37, 42xpcid 17428 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
10096, 99eqtrd 2859 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((Id‘𝑇)‘((1st𝑃)‘𝑥)) = ⟨((Id‘𝐷)‘((1st𝐹)‘𝑥)), ((Id‘𝐸)‘((1st𝐺)‘𝑥))⟩)
10189, 94, 1003eqtr4d 2869 . . . 4 ((𝜑𝑥 ∈ (Base‘𝐶)) → ((𝑥(2nd𝑃)𝑥)‘((Id‘𝐶)‘𝑥)) = ((Id‘𝑇)‘((1st𝑃)‘𝑥)))
102 eqid 2824 . . . . . . 7 (comp‘𝐷) = (comp‘𝐷)
103353ad2ant1 1130 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹)(𝐶 Func 𝐷)(2nd𝐹))
104 simp21 1203 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑥 ∈ (Base‘𝐶))
105 simp22 1204 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑦 ∈ (Base‘𝐶))
106 simp23 1205 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑧 ∈ (Base‘𝐶))
107 simp3l 1198 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦))
108 simp3r 1199 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))
1092, 3, 23, 102, 103, 104, 105, 106, 107, 108funcco 17130 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)))
110 eqid 2824 . . . . . . 7 (comp‘𝐸) = (comp‘𝐸)
11153ad2ant1 1130 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐺 ∈ (𝐶 Func 𝐸))
11238, 111, 39sylancr 590 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺)(𝐶 Func 𝐸)(2nd𝐺))
1132, 3, 23, 110, 112, 104, 105, 106, 107, 108funcco 17130 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓)))
114109, 113opeq12d 4792 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩ = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
11543ad2ant1 1130 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐹 ∈ (𝐶 Func 𝐷))
116273ad2ant1 1130 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → 𝐶 ∈ Cat)
1172, 3, 23, 116, 104, 105, 106, 107, 108catcocl 16945 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥(Hom ‘𝐶)𝑧))
1181, 2, 3, 115, 111, 104, 106, 117prf2 17441 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = ⟨((𝑥(2nd𝐹)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)), ((𝑥(2nd𝐺)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))⟩)
1191, 2, 3, 115, 111, 104prf1 17439 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑥) = ⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩)
1201, 2, 3, 115, 111, 105prf1 17439 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑦) = ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩)
121119, 120opeq12d 4792 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩ = ⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩)
1221, 2, 3, 115, 111, 106prf1 17439 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝑃)‘𝑧) = ⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)
123121, 122oveq12d 7156 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧)) = (⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩))
1241, 2, 3, 115, 111, 105, 106, 108prf2 17441 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝑃)𝑧)‘𝑔) = ⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩)
1251, 2, 3, 115, 111, 104, 105, 107prf2 17441 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑦)‘𝑓) = ⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩)
126123, 124, 125oveq123d 7159 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩))
127363ad2ant1 1130 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐹):(Base‘𝐶)⟶(Base‘𝐷))
128127, 104ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑥) ∈ (Base‘𝐷))
129413ad2ant1 1130 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (1st𝐺):(Base‘𝐶)⟶(Base‘𝐸))
130129, 104ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑥) ∈ (Base‘𝐸))
131127, 105ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑦) ∈ (Base‘𝐷))
132129, 105ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑦) ∈ (Base‘𝐸))
133127, 106ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐹)‘𝑧) ∈ (Base‘𝐷))
134129, 106ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((1st𝐺)‘𝑧) ∈ (Base‘𝐸))
1352, 3, 55, 103, 104, 105funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐹)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
136135, 107ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐹)𝑦)‘𝑓) ∈ (((1st𝐹)‘𝑥)(Hom ‘𝐷)((1st𝐹)‘𝑦)))
1372, 3, 61, 112, 104, 105funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑥(2nd𝐺)𝑦):(𝑥(Hom ‘𝐶)𝑦)⟶(((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
138137, 107ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝐺)𝑦)‘𝑓) ∈ (((1st𝐺)‘𝑥)(Hom ‘𝐸)((1st𝐺)‘𝑦)))
1392, 3, 55, 103, 105, 106funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐹)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
140139, 108ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐹)𝑧)‘𝑔) ∈ (((1st𝐹)‘𝑦)(Hom ‘𝐷)((1st𝐹)‘𝑧)))
1412, 3, 61, 112, 105, 106funcf2 17127 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (𝑦(2nd𝐺)𝑧):(𝑦(Hom ‘𝐶)𝑧)⟶(((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
142141, 108ffvelrnd 6833 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑦(2nd𝐺)𝑧)‘𝑔) ∈ (((1st𝐺)‘𝑦)(Hom ‘𝐸)((1st𝐺)‘𝑧)))
14316, 17, 18, 55, 61, 128, 130, 131, 132, 102, 110, 24, 133, 134, 136, 138, 140, 142xpcco2 17426 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (⟨((𝑦(2nd𝐹)𝑧)‘𝑔), ((𝑦(2nd𝐺)𝑧)‘𝑔)⟩(⟨⟨((1st𝐹)‘𝑥), ((1st𝐺)‘𝑥)⟩, ⟨((1st𝐹)‘𝑦), ((1st𝐺)‘𝑦)⟩⟩(comp‘𝑇)⟨((1st𝐹)‘𝑧), ((1st𝐺)‘𝑧)⟩)⟨((𝑥(2nd𝐹)𝑦)‘𝑓), ((𝑥(2nd𝐺)𝑦)‘𝑓)⟩) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
144126, 143eqtrd 2859 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)) = ⟨(((𝑦(2nd𝐹)𝑧)‘𝑔)(⟨((1st𝐹)‘𝑥), ((1st𝐹)‘𝑦)⟩(comp‘𝐷)((1st𝐹)‘𝑧))((𝑥(2nd𝐹)𝑦)‘𝑓)), (((𝑦(2nd𝐺)𝑧)‘𝑔)(⟨((1st𝐺)‘𝑥), ((1st𝐺)‘𝑦)⟩(comp‘𝐸)((1st𝐺)‘𝑧))((𝑥(2nd𝐺)𝑦)‘𝑓))⟩)
145114, 118, 1443eqtr4d 2869 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶)) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧))) → ((𝑥(2nd𝑃)𝑧)‘(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓)) = (((𝑦(2nd𝑃)𝑧)‘𝑔)(⟨((1st𝑃)‘𝑥), ((1st𝑃)‘𝑦)⟩(comp‘𝑇)((1st𝑃)‘𝑧))((𝑥(2nd𝑃)𝑦)‘𝑓)))
1462, 19, 3, 20, 21, 22, 23, 24, 27, 32, 44, 50, 81, 101, 145isfuncd 17124 . . 3 (𝜑 → (1st𝑃)(𝐶 Func 𝑇)(2nd𝑃))
147 df-br 5048 . . 3 ((1st𝑃)(𝐶 Func 𝑇)(2nd𝑃) ↔ ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
148146, 147sylib 221 . 2 (𝜑 → ⟨(1st𝑃), (2nd𝑃)⟩ ∈ (𝐶 Func 𝑇))
14915, 148eqeltrd 2916 1 (𝜑𝑃 ∈ (𝐶 Func 𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3479  cop 4554   class class class wbr 5047  cmpt 5127   × cxp 5534  Rel wrel 5541   Fn wfn 6331  wf 6332  cfv 6336  (class class class)co 7138  cmpo 7140  1st c1st 7670  2nd c2nd 7671  Basecbs 16472  Hom chom 16565  compcco 16566  Catccat 16924  Idccid 16925   Func cfunc 17113   ×c cxpc 17407   ⟨,⟩F cprf 17410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-4 11688  df-5 11689  df-6 11690  df-7 11691  df-8 11692  df-9 11693  df-n0 11884  df-z 11968  df-dec 12085  df-uz 12230  df-fz 12884  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-hom 16578  df-cco 16579  df-cat 16928  df-cid 16929  df-func 17117  df-xpc 17411  df-prf 17414
This theorem is referenced by:  prf1st  17443  prf2nd  17444  uncfcl  17474  uncf1  17475  uncf2  17476  yonedalem1  17511  yonedalem21  17512  yonedalem22  17517
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